Copyright © 2009 Pearson Addison-Wesley 3.3-1 3 Radian Measure and Circular Functions.

Copyright © 2009 Pearson Addison-Wesley 3.3-1

3Radian Measure and Circular Functions


3.1 Radian Measure

3.2 Applications of Radian Measure

3.3 The Unit Circle and Circular Functions

3.4 Linear and Angular Speed

3Radian Measure and Circular Functions

Copyright © 2009 Pearson Addison-Wesley

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The Unit Circle and Circular Functions

3.3

Circular Functions ▪ Finding Values of Circular Functions ▪ Determining a Number with a Given Circular Function Value ▪ Applying Circular Functions


Circular Functions

A unit circle has its center at the origin and a radius of 1 unit.

The trigonometric functions of angle θ in radians are found by choosing a point (x, y) on the unit circle can be rewritten as functions of the arc length s.

When interpreted this way, they are called circular functions.


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For any real number s represented by a directed arc on the unit circle,

Circular Functions


The Unit Circle


The Unit Circle

The unit circle is symmetric with respect to the x-axis, the y-axis, and the origin.

If a point (a, b) lies on the unit circle, so do (a,–b), (–a, b) and (–a, –b).


The Unit Circle

For a point on the unit circle, its reference arc is the shortest arc from the point itself to the nearest point on the x-axis.

For example, the quadrant I real number

is associated with the point on the

unit circle.


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The Unit Circle

Since sin s = y and cos s = x, we can replace x and y in the equation of the unit circle

to obtain the Pythagorean identity


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Sine and Cosine Functions:

Domains of Circular Functions

Tangent and Secant Functions:

Cotangent and Cosecant Functions:


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Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians.

This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations.

Calculators must be in radian mode when finding circular function values.

Evaluating A Circular Function


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Find the exact values of

Example 1 FINDING EXACT CIRCULAR FUNCTION VALUES

Evaluating a circular function

at the real number is

equivalent to evaluating it at

radians.

An angle of intersects the

circle at the point (0, –1).

Since sin s = y, cos s = x, and


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Use the figure to find the exact values of

Example 2(a) FINDING EXACT CIRCULAR FUNCTION VALUES

The real number

corresponds to the

unit circle point


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Use the figure and the definition of tangent to find

the exact value of

Example 2(b) FINDING EXACT CIRCULAR FUNCTION VALUES

negative directionyields the same ending point as moving around the

Moving around the unit

circle units in the

circle units in the positive direction.


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corresponds to

Example 2(b) FINDING EXACT CIRCULAR FUNCTION VALUES


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Use reference angles and degree/radian

conversion to find the exact value of

Example 2(c) FINDING EXACT CIRCULAR FUNCTION VALUES

An angle of corresponds to an angle of 120°.

In standard position, 120° lies in quadrant II with a reference angle of 60°, so

Cosine is negative in quadrant II.


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Example 3 APPROXIMATING CIRCULAR FUNCTION VALUES

Find a calculator approximation for each circular function value.

(a) cos 1.85 (b) cos .5149≈ –.2756 ≈ .8703


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Example 3 APPROXIMATING CIRCULAR FUNCTION VALUES (continued)

Find a calculator approximation for each circular function value.

(c) cot 1.3209 (d) sec –2.9234≈ .2552 ≈ –1.0243


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Caution

A common error in trigonometry is using a calculator in degree mode when radian mode should be used.

Remember, if you are finding a circular function value of a real number, the calculator must be in radian mode.


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Example 4(a) FINDING A NUMBER GIVEN ITS CIRCULAR FUNCTION VALUE

Approximate the value of s in the interval if cos s = .9685.

Use the inverse cosine function of a calculator.

, so in the given interval, s ≈ .2517.


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Example 4(b) FINDING A NUMBER GIVEN ITS CIRCULAR FUNCTION VALUE

Find the exact value of s in the interval if tan s = 1.

Recall that , and in quadrant III, tan s is negative.


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Example 5 MODELING THE ANGLE OF ELEVATION OF THE SUN

The angle of elevation of the sun in the sky at any latitude L is calculated with the formula

where corresponds to sunrise andoccurs if the sun is directly overhead. ω is the number of radians that Earth has rotated through since noon, when ω = 0. D is the declination of the sun, which varies because Earth is tilted on its axis.


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Example 5 MODELING THE ANGLE OF ELEVATION OF THE SUN (continued)

Sacramento, CA has latitude L = 38.5° or .6720 radian. Find the angle of elevation of the sun θ at 3 P.M. on February 29, 2008, where at that time, D ≈ –.1425 and ω ≈ .7854.


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Example 5 MODELING THE ANGLE OF ELEVATION OF THE SUN (continued)

The angle of elevation of the sun is about .4773 radian or 27.3°.

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Copyright © 2009 Pearson Addison-Wesley 3.3-1 3 Radian Measure and Circular Functions.

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